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Knots, Links, Spatial Graphs, and Algebraic Invariants 689 Knots, Links, Spatial Graphs, and Algebraic Invariants AMS Special Session on Algebraic and Combinatorial Structures in Knot Theory AMS Special Session on Spatial Graphs October 24–25, 2015 California State University, Fullerton, CA Erica Flapan Allison Henrich Aaron Kaestner Sam Nelson Editors American Mathematical Society 689 Knots, Links, Spatial Graphs, and Algebraic Invariants AMS Special Session on Algebraic and Combinatorial Structures in Knot Theory AMS Special Session on Spatial Graphs October 24–25, 2015 California State University, Fullerton, CA Erica Flapan Allison Henrich Aaron Kaestner Sam Nelson Editors American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Dennis DeTurck, Managing Editor Michael Loss Kailash Misra Catherine Yan 2010 Mathematics Subject Classification. Primary 05C10, 57M15, 57M25, 57M27. Library of Congress Cataloging-in-Publication Data Names: Flapan, Erica, 1956- editor. Title: Knots, links, spatial graphs, and algebraic invariants : AMS special session on algebraic and combinatorial structures in knot theory, October 24-25, 2015, California State University, Fullerton, CA : AMS special session on spatial graphs, October 24-25, 2015, California State University, Fullerton, CA / Erica Flapan [and three others], editors. Description: Providence, Rhode Island : American Mathematical Society, [2017] | Series: Con- temporary mathematics ; volume 689 | Includes bibliographical references. Identifiers: LCCN 2016042011 | ISBN 9781470428471 (alk. paper) Subjects: LCSH: Knot theory–Congresses. | Link theory–Congresses. | Graph theory–Congresses. | Invariants–Congresses. | AMS: Combinatorics – Graph theory – Planar graphs; geometric and topological aspects of graph theory. msc | Manifolds and cell complexes – Low-dimensional topology – Relations with graph theory. msc | Manifolds and cell complexes – Low-dimensional topology – Knots and links in S3.msc| Manifolds and cell complexes – Low-dimensional topology – Invariants of knots and 3-manifolds. msc Classification: LCC QA612.2 .K565 2017 | DDC 514/.2242–dc23 LC record available at https://lccn.loc.gov/2016042011 DOI: http://dx.doi.org/10.1090/conm/689 Color graphic policy. Any graphics created in color will be rendered in grayscale for the printed version unless color printing is authorized by the Publisher. In general, color graphics will appear in color in the online version. Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink service. For more information, please visit: http://www.ams.org/rightslink. Send requests for translation rights and licensed reprints to [email protected]. Excluded from these provisions is material for which the author holds copyright. In such cases, requests for permission to reuse or reprint material should be addressed directly to the author(s). Copyright ownership is indicated on the copyright page, or on the lower right-hand corner of the first page of each article within proceedings volumes. c 2017 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 222120191817 Contents Knot Theoretic Structures Preface: Knots, Graphs, Algebra and Combinatorics v The first coefficient of Homflypt and Kauffman polynomials: Vertigan proof of polynomial complexity using dynamic programming Jozef´ H. Przytycki 1 Linear Alexander quandle colorings and the minimum number of colors Mohamed Elhamdadi and Jeremy Kerr 7 Quandle identities and homology W. Edwin Clark and Masahico Saito 23 Ribbonlength of folded ribbon unknots in the plane Elizabeth Denne, Mary Kamp, Rebecca Terry, and Xichen (Catherine) Zhu 37 Checkerboard framings and states of virtual link diagrams Heather A. Dye 53 Virtual covers of links II Micah Chrisman and Aaron Kaestner 65 Spatial Graph Theory Recent developments in spatial graph theory Erica Flapan, Thomas W. Mattman, Blake Mellor, Ramin Naimi, and Ryo Nikkuni 81 Order nine MMIK graphs Thomas W. Mattman, Chris Morris, and Jody Ryker 103 A chord graph constructed from a ribbon surface-link Akio Kawauchi 125 The Kn+5 and K32,1n families and obstructions to n-apex Thomas W. Mattman and Michael Pierce 137 Partially multiplicative biquandles and handlebody-knots Atsushi Ishii and Sam Nelson 159 iii iv CONTENTS Tangle insertion invariants for pseudoknots, singular knots, and rigid vertex spatial graphs Allison Henrich and Louis H. Kauffman 177 Preface: Knots, graphs, algebra & combinatorics Part I: Knot Theoretic Structures The field of knot theory has come a long way since Kurt Reidemeister [19]wrote his groundbreaking book on the subject in 1932. For most of the last century, the focus of research in this area was dominated by classical knot theory, which studies simple closed curves in R3 or S3, perhaps equipped with an orientation or a framing. With the introduction of virtual knot theory in the late 1990’s [14]came an explosion of combinatorial generalizations of knot theory that continues to this day. Each of these combinatorial theories focuses on equivalence classes of diagrams that are preserved under a set of Reidemeister-style moves. Examples include, but are not limited to, the following. • Virtual Knots are defined in terms of knots in thickened orientable sur- faces. Virtual knots and links have diagrams with extra virtual crossings representing genus in the supporting surface of the diagrams [4, 13, 14]. • Twisted Virtual Knots are an extension of virtual knots in which the supporting surface is allowed to be non-orientable [1]. • Classical and Virtual Pseudoknots are equivalence classes of classical or virtual knot diagrams that may be missing certain classical crossing in- formation [9, 10]. • Flat Knots are homotopy classes of virtual knots, equivalent to virtual knots modulo a classical crossing change move [7, 11]. • Free Knots are equivalence classes of flat knots with an extra virtualization move [16]. • Singular Knots are rigid isotopy classes of knotted 4-regular graphs [21]. • Handlebody-Knots are ambient isotopy classes of handlebodies embedded in R3, and are equivalent to embeddings of trivalent graphs in R3 together with an extra move [12]. • Marked Graph Diagrams are knot diagrams with an extra crossing type representing knotted surfaces in R4 [23]. • Ribbon Knots are knots that can be represented by thin strips of paper folded flat in the plane [15]. Many of these combinatorial knot theories have associated algebraic structures such as quandles, biquandles, racks, etc., whose axioms are derived from the appro- priate set of Reidemeister-type moves. We can think of these algebraic structures as colorings of the relevant diagrams, and then count homomorphisms between them to get integer-valued invariants. Such invariants can be enhanced with information from homology theories, such as quandle and biquandle homology [2, 3], to obtain stronger invariants. Recently, such algebraic structures and their homologies have v vi PREFACE: KNOTS, GRAPHS, ALGEBRA & COMBINATORICS been generalized to structures including unital shelves and Yang-Baxter operators [17]. Thefirstpartofthisvolumeisdevotedtocurrentresultsonalgebraic,combina- torial, and geometric structures associated with knot theory and its combinatorial generalizations. On the algebraic side, the volume includes the work of J. Prztycki on knot polynomial complexity, results of M. Elhamdadi and J. Kerr on quandle colorings, and a paper of E. Clark and M. Saito on quandle identities and homology. From the combinatorial and geometric domain, we include a paper by E. Denne, M Kamp, R. Terry, and C. Zhu’s on folded ribbon knots, an article by H. Dye which generalizes classical checkerboard colorings to virtual knots and links, and a paper by M. Chrisman and A. Kaestner which connects virtual knot theory and classical knot theory by using geometric techniques. Part II: Spatial Graph Theory Spatial graph theory developed in the 1980’s when topologists began using the tools of knot theory to study ambient isotopy classes of graphs embedded in R3 or S3. Later, this area came to be known as spatial graph theory to distinguish it from the study of abstract graphs. Much of the current work in spatial graph theory can trace its roots back either to the ground breaking results of John Conway and Cameron Gordon on intrinsic knotting and linking of graphs in S3 or to the topology of non-rigid molecules [8, 20, 22]. In particular, in 1983, Conway and Gordon [6] 3 proved that every embedding of the complete graph K6 in S contains a non-trivial link and every embedding of the complete graph K7 contains a non-trivial knot. 3 Since such links and knots exist for every embedding of the graphs in R , K6 is said to be intrinsically linked and
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